Internal reflection spectroscopy, also known as Attenuated Total Reflectance (ATR) spectroscopy, has been know for many years, and is a widely used method of sampling in infrared (IR) and fluorescence spectroscopy, as well as in other spectroscopies. Mid-wavelength infrared (MWIR), or intermediate infrared (IIR), spectroscopy has over the years become a technique of choice when specificity is of utmost importance. It has historically been a difficult technique to use for several reasons. First, absorptivities of many materials are quite high in the mid-wavelength infrared region of the electromagnetic spectrum (e.g., from about 3-8 μm) While this is good from the standpoint of sensitivity, it makes sampling sometimes complex. As a result, a wide variety of sampling technologies have been developed to help introduce the sample to the spectrometer in an ideal fashion. A ubiquitous and problematic sample component is water. In the near-infrared (NIR) region, using wavelengths from about 800 nm to 2500 nm, another problem that can arise is the fact that the path length may be too short. One advantage is that near-infrared can typically penetrate much farther into a sample than mid infrared radiation.
One problem faced when using spectroscopy is the fact that many sample preparations contain water. Water has a very high absorbance in the mid-infrared. Therefore, in order to measure a spectrum of water in the classical mid-infrared region of 4000-400 cm−1, the path length must be limited to less than a few 10 s of microns. ATR can provide this very small path length needed. In other situations however, the path length of ATR is too small for ideal sampling. This can be the main problem when trying to make measurements through mammalian skin or other biological tissue, or when the desired spectral information is from a deeper depth and not adjacent the surface of the mammalian skin.
Attenuated Total Reflectance (ATR) is often indicated in difficult sampling situations. The spectroscopic usefulness of the effect was first noticed in the 1960s by Fahrenfort and is predictable from basic optical physics. Basically, when light propagates through a medium of high refractive index and approaches an interface with a material of lower refractive index, a transmission and a reflection will occur. The relative strengths of these transmissions and reflections are governed by the Fresnel equations:
                                          r            ⊥                    ≡                                    E              r                                      E              i                                      =                                                                              n                  1                                                  μ                  1                                            ⁢              cos              ⁢                                                          ⁢              θ                        -                                                            n                  2                                                  μ                  2                                            ⁢              cos              ⁢                                                          ⁢                              θ                ′                                                                                                          n                  1                                                  μ                  1                                            ⁢              cos              ⁢                                                          ⁢              θ                        +                                                            n                  2                                                  μ                  2                                            ⁢              cos              ⁢                                                          ⁢                              θ                ′                                                                        (        1        )                                          t          ≡                                    E              t                                      E              i                                      =                              2            ⁢                                          n                1                                            μ                1                                      ⁢            cos            ⁢                                                  ⁢            θ                                                                                n                  1                                                  μ                  1                                            ⁢              cos              ⁢                                                          ⁢              θ                        +                                                            n                  2                                                  μ                  2                                            ⁢              cos              ⁢                                                          ⁢                              θ                ′                                                                        (        2        )                                                      r                                                                        ≡                                    E              r                                      E              i                                      =                                                                              n                  2                                                  μ                  2                                            ⁢              cos              ⁢                                                          ⁢              θ                        -                                                            n                  1                                                  μ                  1                                            ⁢              cos              ⁢                                                          ⁢                              θ                ′                                                                                                          n                  1                                                  μ                  1                                            ⁢              cos              ⁢                                                          ⁢                              θ                ′                                      +                                                            n                  2                                                  μ                  2                                            ⁢              cos              ⁢                                                          ⁢              θ                                                          (        3        )                                                      t                                                                        ≡                                    E              t                                      E              i                                      =                              2            ⁢                                          n                1                                            μ                1                                      ⁢            cos            ⁢                                                  ⁢            θ                                                                                n                  1                                                  μ                  1                                            ⁢              cos              ⁢                                                          ⁢                              θ                ′                                      +                                                            n                  2                                                  μ                  2                                            ⁢              cos              ⁢                                                          ⁢              θ                                                          (        4        )            The Fresnel equations give the ratio of the reflected and transmitted electric field amplitude to initial electric field for electromagnetic radiation incident on a dielectric.
In general, when a wave reaches a boundary between two different dielectric constants, part of the wave is reflected and part is transmitted, with the sum of the energies in these two waves equal to that of the original wave. Examination of these equations reveals that when the light is traversing through a high index medium and approaching an interface with a low index medium, the reflected component can be total, with no light being transmitted. The angle at which this occurs is called the critical angle and is defined by the following equation (5):
                              θ          C                =                              sin                          -              1                                ⁡                      (                                          n                2                                            n                1                                      )                                              (        5        )            
The reflected component has an angle of reflection equal and opposite to the angle of incidence upon the interface. Above the critical angle, all light is reflected. Below the critical angle, some light would transmit through the interface according to the above Fresnel equations. A device operating in this mode would use light that refracts according to Snell's Law (equation (6)):n1 sin θ=n2 sin θ′  (6)
As previously stated, above the critical angle reflection is total. Fahrenfort first noticed that upon total reflection, a standing, or evanescent, wave is set up at the interface between high and low index. The wave has an exponentially decaying intensity into the rarer (lower index) medium. If an absorbing substance is placed in the vicinity of this evanescent (standing) wave, which extends a distance into the rarer medium, it can absorb portions of the light in specific wavelengths corresponding to the absorption properties of the material. In this way, the total reflection is said to be “frustrated” by the absorption of the sample. The returning light at the detector then is evaluated to determine the missing energy. It follows that this mode can be used to obtain an infrared spectrum of a material in contact with the high index medium through which the light is traveling. The strength of this interaction can be predicted through several equations developed by Harrick. First, the depth of penetration is defined as the 1/e point of the exponential decay of the evanescent (standing) wave (equation (7)):
                              d          p                =                              λ            /                          n              1                                            2            ⁢                                          π                ⁡                                  (                                                                                    sin                        2                                            ⁢                      θ                                        -                                                                  (                                                                              n                            2                                                                                n                            1                                                                          )                                            2                                                        )                                                            1                2                                                                        (        7        )            where n2 is the sample refractive index and n1 is the crystal refractive index. The depth of penetration is defined as the point at which the strength of the evanescent wave electric vector decays to a value of 1/e (where e is Euler's number) from its original strength. Quick calculations are often done using the depth of penetration to characterize the strength of signal that will be obtained with ATR. The quick calculations may be less accurate but are suitable for providing a guide. A more accurate equation for the point where the evanescent wave electric vector decays was derived by Harrick, namely the effective thickness or effective depth, de.
An additional complication arises if the sample is thin compared to the 1/e point of the evanescent wave. The effective thickness calculation results in a number that can be used in Beer's Law calculations, and is closely related to the path length in a transmission measurement made at normal incidence. There are now three refractive indices to worry about: n1, the index of the crystal, n2, the index of the thin layer of sample, and n3, the index of whatever is beyond the sample, usually air. Also, since the geometry is usually not near-normal, the calculation must be done for three orthogonal axes. Finally, the measurement is polarization dependent and should be calculated for two orthogonal polarizations. For purposes of this discussion, the thin layer is assumed to by isotropic and the polarization is deemed to be random. So the effective depth equation, for thin layers of sample where the sample layer thickness is much less than the depth of penetration, is as follows:
                              d          e                =                              1                          cos              ⁢                                                          ⁢              θ                                ⁢                                    n              2                                      n              1                                ⁢                                    d              p                        2                    ⁢                                    E              02                              r                ⁢                                                                  ⁢                2                                      ·                          (                                                exp                  ⁡                                      (                                          -                                                                        2                          ⁢                                                      z                            i                                                                                                    d                          p                                                                                      )                                                  -                                  exp                  ⁡                                      (                                          -                                                                        2                          ⁢                                                      z                            f                                                                                                    d                          p                                                                                      )                                                              )                                                          (        8        )            where the z values are the initial and final z-dimension positions of the film relative to the surface of the ATR prism. The E term is the square of the strength of the electric vector in medium 2 E is proportional to light intensity. For polarized incident lightE02,∥r2=E02,xr2+E02,zr2  (9)andE02,⊥r2=E02,yr2  (10)and this results inde,∥=dex+dez  (11)andde,⊥=dey  (12)andde,random=(de,⊥+de,∥)/2  (13)
The three orthogonal electric field components are calculated from Fresnel's equations:
                              E                                    0              ⁢              x                        ,            2                    r                =                              2            ⁢            cos            ⁢                                                  ⁢                                          θ                ⁡                                  (                                                                                    sin                        2                                            ⁢                      θ                                        -                                          n                      31                      2                                                        )                                                            1                /                2                                                                                                          (                                      1                    -                                          n                      31                      2                                                        )                                                  1                  /                  2                                            ⁡                              [                                                                            (                                              1                        +                                                  n                          31                          2                                                                    )                                        ⁢                                          sin                      2                                        ⁢                    θ                                    -                                      n                    31                    2                                                  ]                                                    1              /              2                                                          (        14        )                                                      E                                          0                ⁢                z                            ,              2                        r                    =                                    2              ⁢              cos              ⁢                                                          ⁢              θsinθ              ⁢                                                          ⁢                              n                31                2                                                                                                          (                                          1                      -                                              n                        31                        2                                                              )                                                        1                    /                    2                                                  ⁡                                  [                                                                                    (                                                  1                          +                                                      n                            31                            2                                                                          )                                            ⁢                                              sin                        2                                            ⁢                      θ                                        -                                          n                      31                      2                                                        ]                                                            1                /                2                                                    ⁢                                  ⁢        and                            (        15        )                                          E                                    0              ⁢              y                        ,            2                    r                =                              2            ⁢            cos            ⁢                                                  ⁢            θ                                              (                              1                -                                  n                  31                  2                                            )                                      1              /              2                                                          (        16        )            
In the equations immediately above, a thin film approximation is used, in order to greatly simplify the calculation of the field strength. As previously mentioned, Harrick proposed this approximation. The requirement to use this approximation is that the film must be very thin relative to the depth of penetration if the sample were infinitely thick. The depth of penetration for a thick film at 6 μm measuring wavelength would be 2.32 μm. A monolayer of anthrax spores, for example, would have a thickness of approximately 0.4 μm, so the thin film approximation is valid for early detection and identification of anthrax spores deposited onto an ATR prism. The values used in the above equations are as follows: n1=2.2, n2=1.5, n3=1.0, θ=45°, zi=0. and zf=0.4 μm. Calculated values for the field strength are as follows: E0x,2r=1.37, E0z,2r=0.79, and E0y,2r=1.60. Calculated effective path for each vector are dexiso=0.45 μm, deyiso=0.62 μm, deziso=0.15 μm, de,∥iso=0.60 μm, de,⊥iso=0.62 μm, and de,randomiso=0.61 μm. The final value for effective thickness is therefore 0.61 μm.
A single reflection through the ATR system modeled here would give rise to a signal (at 6 μm wavelength) that is comparable to a layer of spores measured in transmission that is 0.61 μm thick, assuming a spore monolayer with a thickness of 0.4 μm. So the ATR technique, even in a single reflection, gives rise to a spectrum with 1.5× the strength of a transmission measurement. This figure can be increased dramatically by using multiple reflections, making ATR infrared an excellent identifier of biological warfare agents such as anthrax.
Other concepts relating to ATR spectroscopy are disclosed in, for example, U.S. Pat. Nos. 6,908,773 to Li et al. for ATR-FTIR Metal Surface Cleanliness Monitoring; 7,218,270 to Tamburino for ATR Trajectory Tracking System (A-Track); 6,841,792 to Bynum et al. for ATR Crystal Device; 6,493,080 to Boese for ATR Measuring Cell for FTIR Spectroscopy; 6,362,144 to Berman et al. for Cleaning System for Infrared ATR Glucose Measurement System (II); 6,141,100 to Burka et al. for Imaging ATR Spectrometer; 6,430,424 to Berman et al. for Infrared ATR Glucose Measurement System Utilizing a Single Surface of Skin.
An often overlooked benefit of the ATR sampling mode for detecting and classifying samples, however, is the immunity to the effects of scatter. Harrick notes that the ATR mode, unlike transmission or regular reflectance, removes the effect of light scatter. Even if a sample is granular in nature, a situation that normally would give rise to light scattering, the ATR spectrum will maintain a flat baseline. This means that different preparations of the same sample can be more similar to each other, and therefore easier to classify in the same group. If there exists real chemical differences between two samples, the differences are more easily discerned because the sample morphology, preparation, and packing are removed as variables. An advantage of ATR, often overlooked, is its immunity to the effects of scatter. A “perfect” infrared spectrum would contain only information related to the molecular structure of the sample. Sampling artifacts almost always are superimposed on this pure spectrum. However ATR can remove some of the differences due to sample scatter, improving the ability to identify and classify a sample. This can be a huge advantage in the area of tissue spectroscopy.
An interesting recurring theme in the spectroscopy literature is the admonition to stay away from the critical angle (Internal Reflection Spectroscopy: Theory and Applications, Francis M. Mirabella, CRC Press, 1993) because spectral distortions will result. This was noted early on in the seminal book by Harrick, and has been repeated many times since. The basis for this warning is seen in the depth of penetration equations listed above. As the angle of incidence gets smaller and approaches the critical angle, the depth of penetration of the evanescent wave into the rarer medium gets larger and larger, up until the critical angle, at which point the total internal reflection condition no longer holds. Below the critical angle, internal reflectance turns into the much more common and much less useful external reflectance. External reflectance is also governed by the laws of Fresnel reflection, but the resulting reflection is no longer total. In external reflection, it is not possible to couple a large efficiency of energy back into the ATR prism and subsequently to the detector.
For many samples, it would be desirable to have a large depth of penetration into the sample. This could be achieved by introducing electromagnetic energy very close to a critical angle for the sample. In most spectrometers, the light beam has a significant angular dispersion, in order to fill the detector and obtain high signal-to-noise ratio (SNR). However, because there is much angular dispersion, as the critical angle is approached, a portion of the beam starts to exceed the critical angle, while another portion of the beam is still at an angle that is well away from the critical angle. In addition, in most samples there is dispersion in the refractive index across the spectral region of interest, and so the critical angle is different for different wavelengths. So these factors require the average angle to often be several degrees away from the critical angle.
It can be readily seen that the depth of penetration into the rarer medium can actually become quite large. There are many applications in which a larger depth of penetration would be desirable. The non-invasive measurement of body constituents is amongst these. The teaching, repeated many times in the literature, is that ATR can not have a large path length and can not have a large depth of penetration, because distortions of the spectrum occur near the critical angle. This problem could be overcome by the use of a highly collimated beam of light. Light sources are now available that can be highly collimated, yet still contain excellent amounts of energy. Many lasers such as quantum cascade lasers and light emitting diode (LED) sources are now available that can be highly collimated and still contain large amounts of energy. But this is not a complete solution to the problem.
Another problem that needs to be overcome is the fact that most samples themselves exhibit wavelength dispersion in their refractive index. If useful spectroscopic information about a sample is desired, whether by fluorescence, near infrared, terahertz, or some other spectroscopy, the signal should be collected over some range of wavelengths. It will almost certainly be true that over the wavelength range of interest, the critical angle will vary with wavelength. The critical angle will even change within the same sample depending on various characteristics of the sample, such as the sample morphology or the physical state of the sample. Therefore it is very difficult, if not impossible to know, a priori, where the critical angle will lie, for a given sample at a given wavelength. What is needed is an added dimension to the ATR measurement, namely that of a mapping of not only intensity versus wavelength, but of intensity versus wavelength versus angle of incidence and/or reflection.
An ATR sampler can be designed that allows for multiple reflections. Multiple reflections thereby multiply the strength of the infrared spectrum. The number of reflections can be adjusted to arrive at an optimum effective path length to give the highest possible signal-to-noise ratio. The apparatuses and methods described here provides for measurements that are at least one, and probably two, orders of magnitude more sensitive than making the measurement in a transmission mode or a traditional ATR mode. In order to successfully map the angular space of interest, it would be desirable to cross over the critical angle and also collect data below the critical angle. This data could be useful in determining a true critical angle for each wavelength.